Finite Group Databasebeta

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Cayley diagram

C2×#8269 #183343

order512
nilpotency class3
isoclinismisoclinic class of #468
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4

Aliases

direct productC2×#8269

Subgroups

subgroupG/H
centerC2×C42C2×D4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C4×C4⋊C4
prime radicalC24#117
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

886 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(12,16,18,21)(13,19)(17,24,25,23)(20,26);(3,4,5,8,6,7,9,10)(11,12)(13,17)(14,21)(15,18)(16,22)(19,25)(20,24)(23,26);(4,7)(8,10)(11,13,14,20,15,19,22,26)(12,17,21,24,18,25,16,23)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#183351, C2×#128572
maximal normal subgroup ofC2×#183351, C2×#128572

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90539);